Soft and hard confinement of a two-electron quantum system

A model physical problem is studied in which a system of two electrons is subject either to soft confinement by means of attractive oscillator potentials or by entrapment within an impenetrable spherical box of finite radius $R.$ When hard confinement is present the oscillators can be switched off. Exact analytical solutions are found for special parameter sets, and highly accurate numerical solutions (18 decimal places) are obtained for general cases. Some interesting degeneracy questions are discussed at length.